1. Warm Up Find AB. 1. A(0, 15), B(17, 0) 2. A(–4, 2), B(4, –2)
Solve each equation. Round to the nearest tenth or nearest degree.
6.3
3.1
38°
50°

2. Objectives Find the magnitude and direction of a vector.
Use vectors and vector addition to solve real world problems.

3. The speed and direction an object moves can be represented by a vector
The speed and direction an object moves can be represented by a vector. A vector is a quantity that has both length and direction.
You can think of a vector as a directed line segment. The vector below may be named

4. A vector can also be named using component form.
The component form <x, y> of a vector lists the
horizontal and vertical change from the initial point to the terminal point. The component form of
is <2, 3>.

5. Example 1A: Writing Vectors in Component Form
Write the vector in component form.
The horizontal change from H to G is –3 units.
The vertical change from H to G is 5 units.
So the component form of is <–3, 5>.

6. Example 1B: Writing Vectors in Component Form
Write the vector in component form.
with M(–8, 1) and N(2, –7)
Subtract the coordinates of the initial point from the coordinates of the terminal point.
= <x2 – x1, y2 – y1>
Substitute the coordinates of the given points.
= <2 – (–8), –7 – 1>
= <10, –8>
Simplify.

7. The magnitude of a vector is its length
The magnitude of a vector is its length. The magnitude of a vector is written
When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed. For example, if a vector represents the course a kayaker paddles, the magnitude of the vector is the kayaker’s speed.

8. Example 2: Finding the Magnitude of a Vector
Draw the vector <–1, 5> on a coordinate plane. Find its magnitude to the nearest tenth.
Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (–1, 5) is the terminal point.
Step 2 Find the magnitude. Use the Distance Formula.

9. The direction of a vector is the angle that it makes with a horizontal line. This angle is measured counterclockwise from the positive x-axis. The direction of
is 60°.
The direction of a vector can also be given as a bearing relative to the compass directions north, south, east, and west has a bearing of N 30° E.

10. Example 3: Finding the Direction of a Vector
The force exerted by a skier is given by the vector <1, 4>. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree.
Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. B
Step 2 Find the direction.
Draw right triangle ABC as shown. A is the angle formed by the vector and the x-axis, and A C
So mA = tan-1(4)  76°.

11. Step 2 Find the direction.
Check It Out! Example 3
The force exerted by a tugboat is given by the vector <7, 3>. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree.
Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. B
Step 2 Find the direction.
Draw right triangle ABC as shown. A is the angle formed by the vector and the x-axis, and A C

12. Two vectors are equal vectors if they have the same
magnitude and the same direction. For example,
. Equal vectors do not have to have the same initial point and terminal point.

13. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, Equal vectors are always parallel vectors.

14. Example 4: Identifying Equal and Parallel Vectors
Identify each of the following.
A. equal vectors
Identify vectors with the same magnitude and direction.
B. parallel vectors
Identify vectors with the same or opposite directions.

15. Identify each of the following.
Check It Out! Example 4
Identify each of the following.
a. equal vectors
Identify vectors with the same magnitude and direction.
b. parallel vectors
Identify vectors with the same or opposite directions.

16. Lesson Quiz: Part I
Round angles to the nearest degree and other values to the nearest tenth.
1. Write with S(–5, 2) and T(8, –4) in component form.
2. Write with magnitude 12 and direction 36° in component form.
3. Find the magnitude and direction of the vector <4, 5>.
<13, –6>
<9.7, 7.1>
6.4; 51°

17. Lesson Quiz: Part II
4. Find the sum of the vectors <2, –4> and
<3, 6>. Then find the magnitude and direction of the resultant vector.
5. A boat is heading due east at a constant speed of 35 mi/h. There is an 8 mi/h current moving north. What is the boat’s actual speed and
direction?
<5, 2>; 5.4; 22°
35.9 mi/h; N 77° E